L(s) = 1 | + 2-s − 3.34·3-s + 4-s − 1.21·5-s − 3.34·6-s + 1.21·7-s + 8-s + 8.16·9-s − 1.21·10-s − 1.48·11-s − 3.34·12-s − 2.69·13-s + 1.21·14-s + 4.07·15-s + 16-s + 2.46·17-s + 8.16·18-s + 6.34·19-s − 1.21·20-s − 4.05·21-s − 1.48·22-s − 0.258·23-s − 3.34·24-s − 3.51·25-s − 2.69·26-s − 17.2·27-s + 1.21·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.92·3-s + 0.5·4-s − 0.545·5-s − 1.36·6-s + 0.458·7-s + 0.353·8-s + 2.72·9-s − 0.385·10-s − 0.446·11-s − 0.964·12-s − 0.748·13-s + 0.323·14-s + 1.05·15-s + 0.250·16-s + 0.597·17-s + 1.92·18-s + 1.45·19-s − 0.272·20-s − 0.883·21-s − 0.315·22-s − 0.0539·23-s − 0.682·24-s − 0.702·25-s − 0.529·26-s − 3.32·27-s + 0.229·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.286122441\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.286122441\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 2003 | \( 1 + T \) |
good | 3 | \( 1 + 3.34T + 3T^{2} \) |
| 5 | \( 1 + 1.21T + 5T^{2} \) |
| 7 | \( 1 - 1.21T + 7T^{2} \) |
| 11 | \( 1 + 1.48T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 17 | \( 1 - 2.46T + 17T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 + 0.258T + 23T^{2} \) |
| 29 | \( 1 + 0.940T + 29T^{2} \) |
| 31 | \( 1 - 5.09T + 31T^{2} \) |
| 37 | \( 1 - 0.621T + 37T^{2} \) |
| 41 | \( 1 - 2.70T + 41T^{2} \) |
| 43 | \( 1 + 0.396T + 43T^{2} \) |
| 47 | \( 1 + 6.14T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 - 2.37T + 59T^{2} \) |
| 61 | \( 1 + 0.765T + 61T^{2} \) |
| 67 | \( 1 + 6.41T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + 8.44T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 4.34T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.82481665172656496196116710281, −7.69266275152150740205146185521, −6.75967049498568594861323622436, −6.08852747107929065495770880484, −5.24920058575337607378045100391, −4.97677612288243887600991075486, −4.19664043791731872835276968682, −3.20794863614035169019201863233, −1.76516402852378178382768757144, −0.65768817618720536038000063112,
0.65768817618720536038000063112, 1.76516402852378178382768757144, 3.20794863614035169019201863233, 4.19664043791731872835276968682, 4.97677612288243887600991075486, 5.24920058575337607378045100391, 6.08852747107929065495770880484, 6.75967049498568594861323622436, 7.69266275152150740205146185521, 7.82481665172656496196116710281